The first Crusade was done because the Romans wanted help to defeat the Turks. However it seems the Crusaders had their own ideas. Can anyone tell me if there were positive to the Romans in one way or another and how?
The first Crusade actually liberated some cities for the Byzantines so they did gain some land. Originally the Crusaders agreed to give all conquered territory to the Byzantines but after the first couple of cities fell they reneged on that promise (the Byzantines also failed to support the Crusaders with an army that they had promised so both sides are slightly to blame for this). That being said some cities were handed over to the Byzantines. Tha being said, some of the crusaders also had a bad habit of pillaging land they travelled through so some Byzantine territory was weakened by the Crusaders as they passed through (the peasant crusaders were some of the worst offenders of this and they were also largely ineffective against everyone they fought so they generally just hurt the Empire).
The First Crusade probably brought the most material benefit to the Byzantines. They were directly responsible for reconquering Nicaea, though the event is often cited as a cause of friction between the Byzantines and Crusaders (The Byzantines used subterfuge to get access to the city in the end and were wary of Crusaders entering it). And they soon after won a decisive battle against the Seljuks at Dorylaion. This gave the Byzantines breathing room to reestablish control over Bithynia and western Asia Minor, which had been taken over by minor Turkish princelings and pirates in recent times. Future crusades were not very successful or more focused on supporting the Crusader States rather than the Byzantines. And we all know the aftermath of the Fourth Crusade.
The Crusades also greatly hindered the Byzantines by conquering Antioch, which thanks to geography was a nearly impregnable fortress (the Crusaders capturing it was quite a feat and they nearly failed). Worse yet, it was ruled by Bohemund, a Norman prince of Sicily who only a decade before had tried to conquer the Empire for his father, Roger Guiscard. Since the late 10th century, Antioch had been the lynchpin of Byzantine control over Syria and eastern Anatolia and they would spend the rest of the 12th century trying to reassert control with minimal success.
It's important to note that Crusader armies were typically not very disciplined and accompanied by myriads of camp followers. The Byzantines would basically try to escort them through Europe to prevent them from looting and pillaging "allied" land and then dump them in Anatolia to fend for themselves.
fourth crusade was the worse, but it was already weakened by the previous infighting in the Komnenoi family, and later the Angeloi family but also a bulgarian revolt also helped destabilise the empire.
The fourth crusade was originally to attack the muslims in egypt however, due to a lack of funding and a byzantine emperor asking for support to retake the throne, the fourth crusade divided the lands that were previously held by the byzantines